# Co-ordinate Systems

If we want to think about things geometrically, we need a way to describe where things are in space, where things are in relation to each other and how big they are.

To do that, we can use a co-ordinate system. Linear co-ordinate systems have two properties - units and dimensions. The dimensions correspond to how many different combinations you can make with positions in your space. We will often see dimensions referred to by the canonical variables $x$, $y$, $z$ and so on.

There exist infinitely many points in the other corresponding dimensions for a single point on one dimension. For instance, if you had a two dimensional space and you held $x$ constant at $x=1$, there are still infinitely many points you could pick on the $y$ dimension. If you had a three dimensional space and held $x$ at $x=1$, there are infinitely many points that you could choose in the $y$ dimension, and then after that, there are infinitely many points you could choose in the $z$ dimension.

In the first dimension you would just have a number line made up of every possible point:

In two dimensional space, you have a co-ordinate plane made up of every possible line:

In three dimensional space, you have a volume made up of every possible plane:

Dimensions above the fourth are a little tricky to visualize, but the pattern continues. If two-dimensional space is a plane consisting of consists of every possible line and three-dimensional space is a volume consisting of every possible plane, then think about what that means for four-dimensional space. Or even five-dimensional space.

The logic will generalize to an n-dimensional space

For the sake of simplicitly, we will assume that all co-ordinate systems use the same units, meaning that movement of one step along one dimension and that if you rotated a system such that the one dimension became another, the steps would correspond.